In 1 Corinthians 6, Paul harshly reprimands the Christians at Corinth for publicly taking fellow Christians before a non-Christian court or judge over various alleged wrongs, rather than settling such matters privately between themselves, or within the Christian community.
Paul says that their willingness to do this is itself an indication that they are engaging in activities of which God disapproves (such as defrauding others), and that those who do such things will not inherit the Kingdom of God (meaning, in my opinion, that those who practice such actions in a continuing or unrepentant manner will be indicating that they are not saved -- regardless of whatever profession of Christian faith they may make with their lips).
He then expands upon this subject by listing other practices that, if engaged in in an ongoing, unrepentant manner, with no recognition of the action's wrongfulness, no confession of it, and no attempted forsaking of it, indicate a heart that has not been regenerated by God, and a person who is not saved (irrespective of outward professions of faith). These include:
sexual immorality
idolatry
adultery
homosexual acts
theft
greed
drunkenness
slander
swindling.
(Note that homosexual acts are not singled out here for special condemnation, but are named on an equal basis with other acts that Christians are to avoid. However, each of these behaviors, practiced unrepentingly, with no recognition of wrongfulness, and no indication of awareness of a need to change, give evidence of an unsaved state.)
Paul then notes that some of the Christians at Corinth had in fact practiced such actions prior to receiving Christ. But their acceptance of Christ (accompanied, apparently, by their intent to reform their lives with the help of the Holy Spirit) had resulted in their washing, cleansing, and sanctification.
This would indicate to me that, regardless of whether one believes that homosexuality is an orientation that a person is born with; or is produced by environmental factors over which the individual has no control; or is a conscious choice on the part of the individual, the key with respect to salvation is to avoid expressing that orientation or making that choice by unrepentingly engaging in homosexual acts with others, or (in my opinion, as far as Christians are concerned) by living in a way or relationship (even if no sexual activity is present) that gives the appearance of sexual immorality to others (just as I would also say that it would be wrong for an unmarried heterosexual couple to live together out of wedlock, even if no sexual activity were present).
Also, such actions are never something that Christians should accept or be complacent about, but that they should recognize as contrary to God's commands; that they should consciously seek His help in avoiding; and of which they should repent, and for which they should seek His forgiveness and restoration, if or when they do succumb to them.
If those conditions are met, I would say that it is certainly possible for someone who feels sexual attraction to individuals of the same gender to be saved, just as is the case with individuals who commit the other acts that Paul lists.

The key is to recognize the sin, repent and turn to God to help overcome it.
"Idolatry" can be when your job/money/success come first in your life--anything other than God.
So that list covers just about anything we are willing to put before God in our life. And if that's our sexual sin, that means that sin is more important to us than God. Repentance means putting God first in your life.
Anyone with those sins can be saved! But you clearly are NOT saved if you live a lifestyle in which any of those things is more important to you than God.
Committing any of these sins and then repenting is not the same thing as living a lifestyle that promotes it.